
Though the femto AP could support indoor users, it also causes
interference problems. In this chapter, we proposed a power control
algorithm for the femto AP that will help to reduce the
interference. The proposed scheme reduces the transmission power when
the traffic density of the femto AP is low, so that the femto AP could
avoid any throughput loss. In order to evaluate the performances of
the proposed scheme, we use a finite state markov chain (FSMC)
model. Then, we show some numerical results of the proposed scheme.

\section{Overview and Related works}
In order to solve indoor coverage problem, some research groups such
as the 3GPP and the IEEE 802.16 start to consider femto cell topology
\cite{CAG08FNS,CHS08OFC,3GPP} which means a very small cell structure
in cellular networks. Fig.\ref{femto} shows an example of the
femto cell system. As you can see in the figure, a small femto AP is
used to support indoor users. Since the femto APs are deployed in
indoor areas, they could have good channel
quality to support high-speed packet transmissions.

%================================================================
\begin{figure}
\centering{\includegraphics[width=0.6\columnwidth]{fig/femtomodel}}
\caption{Network configuration of femto cell system} \label{femto}
\end{figure}
%================================================================

Although the femto cell solves the coverage problem involving indoor
users, there are some remained challenges. One of the main issues is
interference problems \cite{CAG08FNS,CHS08OFC,3GPP}. Since many
service providers do not have enough frequency resource to provide
services, femto cells might use the same channel as that of macro
cells. Consequently, when a macro user exists nearby a femto AP, the
femto AP's transmission signals interfere with the receiving signal of
the macro user. Therefore, in downlink case, due to the signal which
is transmitted by femto AP, the data rate of macro user is reduced
because the SINR becomes lower because of the interference. It is a
critical problem from the veiwpoint of service providers since they
want to maintain the performance of macro users. Moreover, since the
femto AP might allow only notified users to connect so that only the
owner of the femto AP could get benefits from the femto AP,
non-authorized users nearby the femto AP can not change their serving
BS and suffer from significant interference from the femto AP even if
the received signal power of the femto AP is larger than that of their
current serving BS. Consequently, in some area, macro users can not
decode any message from their serving base station. This area is
called the "coverage hole". Furthermore, there are not only
interference issues between a femto cell and a macro cell but also
inter femto cell interference problems. Since a femto AP can be easily
deployed by any users, it is hard to control the interference among
various femto APs, particularly when they are installed nearby.

In some previous research work made to solve the above-mentioned
issues, the power control scheme was exploited to reduce interference
\cite{CHS08OFC}, because if a femto AP reduces its transmitted power,
the power changing could reduce interference for neighboring users. In
\cite{CHS08OFC}, when a new femto AP is turned on, the femto AP
estimates received macro cell power and determines its transmission
power to limit interference. Therefore, the power control scheme which
was proposed by \cite{CHS08OFC} guaranteed that a large coverage hole
could be avoided.

If we assume that the incoming traffic rate of a femto AP is lower
than the channel capacity of the femto AP, we could reduce femto AP's
power without any throughput degradation. Fortunately, this assumption
is possible for some cases. For example, because there might be one or
two users in femto cell when the femto AP is deployed in homes, the
femto AP does not need high data rate if the users do not demand lots
of packets. Moreover, backhaul link capacity could be less than air
medium capacity in real environments \cite{CAG08FNS}. For example,
when users of the femto cell watch IPTV or download some large size
data to their PC, the femto AP experiences backhaul bottleneck since
it shares Ethernet link with other services. In that case, the femto
AP does not need to operate with full transmission power. Therefore,
in this chapter, we propose a power control scheme that takes into
consideration the traffic density of the femto AP.

The remaining of this chapter is organized as
follows. Section~\ref{sec:power-contr-accord} defines the system
models. The proposed scheme which is a power control algorithm for a
femto AP is explained in section~\ref{sec:system-model}. In
section~\ref{sec:performance-analysis}, we explain a finite state
Markov chain (FSMC) model to analyze system performance. By using the
FSMC model, numerical results are derived and explained in
section~\ref{sec:numerical-results}. Finally, we conclude this chapter
in section~\ref{sec:conclusions-1}.

\hfill

\section{Power control according to traffic density}\label{sec:power-contr-accord}
\subsection{Packet based power control}
In AWGN wireless channel model, the $C_{ij}$ which is the capacity of the link between node i and node j is expressed as 
%===========================================================
\begin{equation}
\label{capacityeq}
C_{ij}=W \log_{2}(1+\frac{|h_{ii}|P_{i}}{N_{0}+\sum_{j\neq i}|h_{ji}|P_{j}})
\end{equation}
%===========================================================
where $h_{ij}$ refers channel gain between node i and node j and
$P_{i}$ is transmission power of node i. From Eq.(\ref{capacityeq}),
we already know that its own capacity decreases but capcities of other
links increase as its transmission power decreases. Thus, power
control is very useful for interference reduction though its capacity
should be degraded.

When we consider the power control for a femto AP, there are some
issues to solve. One of that is a handoff problem. In general, when
received signal power of a mobile node from one of its neighbor base
stations is higher than the received signal strength form the serving
base staion, the mobile node changes its serving base station to the
base station transmiting the signal which has the highest received
signal strength. So, as the femto AP reduces its transmission power,
mobile terminals which are served by the femto AP should try handoff
more frequently. Additionally, channel variation of neighbor nodes is
also an problem which is occured by the power control. Because channel
fluctuation could make significant error when they use adaptive
channel coding scheme for channel variation, power control scheme of
the femto AP might lead throughput degradation of neighbor
cells. Therefore, we should consider the problems to sugguest a power
control algorithm for femto AP.

%================================================================
\begin{figure}[!t]
\centering{\includegraphics[width=0.5\columnwidth]{fig/algorithm}}
\caption{Relationship between queue length and operation mode} \label{algo}
\end{figure}
%================================================================

To construct a power control algorithm that adapatively operates based
on traffic density, we should think about how to estimate the traffic
density. In our scheme, the queue length of the femto AP is exploited
to estimate the traffic density. Since the queue length implies the
femto AP's required service amount, it is reasonable that the power
control scheme uses the queue length for estimating the traffic
density. In order to reduce the channel fluctuation of neighbor nodes
by the femto AP's power control, transmission power level should
not be changed frequently. Therefore, the femto AP does not check its queue
length every time but checks every $N_L$ frame interval, so that
neighbor nodes could adapt the channel variation. When the femto AP
checks its queue length, the increase in the queue length means that
the serving data rate is less than the input data rate. On the other
hand, the decrease in the queue length means that the incoming data
rate of the femto AP is less than the serving data rate. Therefore,
when the queue length becomes larger, we should increase transmission
power to boost packet transmission speed and when the queue length is
reduced, we could decrease transmission power to degrade the
interference for neighbor users.

To provide a such power control algorithm, the proposed scheme exploits
two operation modes, which will be called mode1 and mode2. If a femto
AP operates in mode1, the femto AP uses half of its power to transmit
and the mode2 means that the femto AP transmits packets with full
power. By using the operation mode and some thresholds which are
$\alpha$ and $\beta$ in the queue length, the femto AP operates in
mode1 if the queue length is less than $\beta$ and in mode2 if the
queue length is larger than $\alpha$ in our proposed
scheme. Becasue $\alpha$ is larger than $\beta$, the proposed
scheme is expected to reduce interference to
neighbors. Moreover, the femto AP broadcasts its operation mode every
frame to prevent unwanted handoff. When a node receives the mode
message, the node can estimate received signal strength of the femto
AP when transmiting with full power. Therefore, by using the
information, the unwanted handover problem should be solved since the
received signal strength of the femto AP could be revised by the
message. Fig.\ref{algo} represents the operation of the proposed
scheme where $P_f$ is the transmission power of the femto AP.


\hfill


\section{System model}\label{sec:system-model}
%====================================================
\begin{table}[!t]
\renewcommand{\arraystretch}{1.2}
\caption{MCS levels in the IEEE 802.16 system}
\label{mcstable}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
MCS level & packets/frame & AMC mode & SINR thrshold value\\ 
\hline
\hline
1 & 2 & QPSK-1/2 & 6.0 dB\\
\hline
2 & 3 & QPSK-3/4 & 9.0 dB\\
\hline
3 & 4 & 16QAM-1/2 & 12.0 dB\\
\hline
4 & 6 & 16QAM-3/4 & 15.0 dB\\
\hline
5 & 9 & 64QAM-3/4 & 21.0 dB\\
\hline
\end{tabular}
\end{center}
\end{table}
%=====================================================
Our proposed algorithm is based on the IEEE 802.16e standard \cite{IEEE} which is developed to provide high-speed data packet services. However, our proposed scheme might be suitable for any other standards. The IEEE 802.16e system exploits adaptive modulation and coding (AMC) schemes to get throughput enhancement \cite{wimax, dhcho}. Generally, since the AMC schemes change modulation and coding scheme (MCS) level by channel state, a base station which wants to transmit some packets by using the AMC scheme should know channel quality indicator (CQI) such as SINR at its destination node. Therefore, the destination node sends CQI value to the base station and the base station determines its MCS level according to the SINR. In the case of the IEEE 802.16e standard, a mobile terminal calculates its SINR value by a preamble signal \cite{IEEE, dhcho}. After the CQI reporting, the base station makes a decision about MCS level by using SINR-MCS table. Table \ref{mcstable} is an example for the SINR-MCS table in the IEEE 802.16 standard. In this chapter, the Table.\ref{mcstable} is used for our modeling and analysis.

We assume that every packet size is the same and packet arrival distribution of a base station follows Poisson distribution. Moreover, we assume that the packet arrival distribution is i.i.d for every frame. Let $A_n$ denote the number of arrival packet at the n-th frame. Then, the distribution of $A_n$ can be represented as
%=============================================================================
\begin{equation}
Pr\{A_{n} = k\}=\frac{\lambda^k e^{-\lambda}}{k!}
\end{equation}
%=============================================================================
where $\lambda$ is the average number of arrival packets at a frame and $k \ge 0$. When the packets arrive, they are stacked on the base station's queue. The queue length is finite and we denote the length of the queue as $Q_{L}$. It means that a base station can't have more than $Q_{L}$ packets at its queue. Therefore, when packets arrive at a base station and the queue of the base station overflows, incoming packets should be dropped.

The Nakagami-m model is used for our time-varying channel model. Then, the probability density function (pdf) of the received SINR $\gamma$ is described as
%=============================================================================
\begin{equation}
p_{\gamma}(\gamma) = \frac{m^m \gamma^{m-1}}{\bar{\gamma}\Gamma (m)}e^{- \frac{m \gamma}{\bar{\gamma}}}
\end{equation}
%=============================================================================
where $\bar{\gamma}$ is an average SINR and $\Gamma (x)$ is a gamma function \cite{smith}. However, we don't consider shadowing. For channel modeling, we assume that the channel state is invariant during a frame transmission and that it can be changed frame-by-frame, since indoor channels might be slowly-varying channels. Thus, we can construct a finite state markov chain (FSMC) channel model \cite{LZ05QAM}. And then, we assume that the femto AP knows perfect CQI at the receiver.


\hfill



\section{Performance analysis}\label{sec:performance-analysis}

In \cite{LZ05QAM}, the authors make a queueing modeling for the MCS
scheme. Similar to \cite{LZ05QAM}, we can make an FSMC model for our
proposed scheme.


%=================================================================
\begin{table}[!t]
\renewcommand{\arraystretch}{1.2}
\caption{MCS level and data rate for channel state}
\label{chantab}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
State              &1&2&3&4&5&6&7&8&9\\ 
\hline
MCS &               0&0&1&2&3&4&4&5&5  \\
\hline
Packets/frame&      0&0&2&3&4&6&6&9&9 \\
\hline
\end{tabular}
\end{center}
\end{table}
%=================================================================
%================================================================
\begin{figure}
\centering{\includegraphics[width=0.4\columnwidth]{fig/channelstate}}
\caption{Channel state transition according to mode change} \label{chanalstate}
\end{figure}
%================================================================
As we mentioned on the system model section, we consider the
Nakagami-m fading channel as wireless channel and we use an FSMC
channel model for performance analysis. In order to make the FSMC
channel, we have to define channel states. For convenient analysis, we
make each channel state have the same MCS level in each SINR range,
e.g. if the SINR range of a channel state is from 6dB to 9dB, the
channel could have the same MCS level 1 from
Table. \ref{mcstable}. Moreover, since our proposed scheme changes the
femto AP's transmission power by 3dB according to the queue length of
the femto AP, we set the SINR distance between adjacent channel states
to 3 dB, so that a channel state changes to a particular state when
the femto AP changes its transmission power. Fig.  \ref{chanalstate}
represents channel states of our modeling. Each of the channel states
has the same MCS level as shown in Table. \ref{chantab} and changes to
a particular channel state when the femto AP changes the transmission
power. The channel state transition probabilities can be obtained by
using the equations in \cite{LZ05QAM}.
%================================================================
\begin{figure}
\centering{\includegraphics[width=0.95\columnwidth]{fig/queuestate}}
\caption{Relation between queue length, channel state and arrival packets.} \label{asdf}
\end{figure}
%================================================================

Let $Q_{n}$, $C_{n}$ and $A_{n}$ denote the number of packets in the
queue, the channel state and the number of arrival packets at time $n$,
respectively, and let $D(C_{n})$ denote the value of packets/frame of
the channel state $C_{n}$. The value of packets/frame can be found in
Table. \ref{chantab}. As shown in Fig. \ref{asdf}, during $n$-th frame,
the femto AP sends $D(C_{n})$ packets from $Q_{n}$ packets in the queue at time $n$
and $A_{n}$ packets arrive in the queue. Consequently, the
relationship between $Q_{n}$ and $Q_{n-1}$ can be represented as
%=============================================================================
\begin{equation}
Q_{n} = min\{ Q_{L}, max\{ 0, Q_{n-1} - D(C_{n-1}) \}+A_{n-1} \}.
\end{equation}
%=============================================================================
Note that, during the $n$-th frame, the number of the transmitted
packets have to be less than $Q_n$ and the number of packets can't
exceed the maximum queue length $Q_{L}$.

We exploit an augmented FSMC to analyze the system \cite{LZ05QAM}. At
first, we think about the FSMC model without the power control
scheme. The state of the FSMC model is defined as a state pair
$(Q_{n},C_{n})$. Let the $P_{\alpha, \beta}$ denote the probability of
the state transition from state $\alpha$ to state $\beta$. For all
$(q,c)\in (Q_{n},C_{n})$ and $(q',c')\in (Q_{n+1},C_{n+1})$,
$P_{(q,c),(q',c')}$ is defined as
%=============================================================================
\begin{eqnarray}
P_{(q,c),(q',c')} &=& 
P(C_{n+1} = c' | C_{n} = c)P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c).
\label{eqadfadif}
\end{eqnarray}
%=============================================================================
In (\ref{eqadfadif}), the $P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c)$ is independent of power level of the femto AP. The $P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c)$ can be calculated as followings:
\begin{compactenum}[\em i)]
\item if $max\{ 0, q-D(c) \} \le q' < Q_{L},$ then $P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c) =P(A_{n} = q' - max\{ 0, q-D(c) \}),$

\item if $0 \le q' < max\{ 0, q-D(c) \},$ then, $P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c) = 0,$

\item if $q'=Q_{L},$ then, $P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c) = 1- \sum_{0 \le q' < Q_{L}} P(Q_{n+1} = q' | Q_{n} = q, C_{n} = c),$

\end{compactenum} 

However, the $P(C_{n+1} = c' | C_{n} = c)$ in (\ref{eqadfadif}), which
is the channel state transition probability, is changed as the average
SINR is changed, since the probability is dependent on the average
SINR \cite{LZ05QAM}. Therefore, we have to make two state transition
matrices for each mode because the femto AP has two level of
transmission power. Let $\bm{P_{1}}$ and $\bm{P_{2}}$ denote the state
transition matrices for mode1 and mode2, respectively. Since the femto
AP uses half power to transmit packets in the case of mode 1, the
$\bm{P_{1}}$ can be constructed by Eq.(\ref{eqadfadif}) where the
average SINR is $\bar{\gamma}-3 (dB)$. Simillary, The $\bm{P_{1}}$ is
also calculated by Eq.(\ref{eqadfadif}) where the average SINR is
$\bar{\gamma} (dB)$.

%================================================================
\begin{figure}[!t]
\centering{\includegraphics[width=0.8\columnwidth]{fig/frameblock}}
\caption{Frame and frame block} \label{frameblock}
\end{figure}
%================================================================

We need a stationary probability which contains the probability about
modes, so that we could analyze system performances based on power
consumption and everage packet drop rate. Therefore, we have to
redefine the state pair to contain the mode state. According to our
proposed scheme, the femto AP changes its transmission power in
$N_{L}$ frames interval. So, we make frame block concept in order to
define the state pair. As shown in Fig.\ref{frameblock}, $N_L$ frames
make one frame block. Let $M_{N}$ denote the mode number at the N-th
frame block. Then, the state pair can be constructed as $(M_{N}, Q_{N
  \times N_{L}}, C_{N \times N_{L}})$. By using the state pair for
frame block, we can make an FSMC model.

Since the operation mode is not changed during a frame block, the
queue length and the channel state are changed by the transition
matrices $\bm{P_{1}}$ during the frame block when the operation mode
is mode1. Simillary, if the operation mode is mode 2, the queue length
and the channel state are changed by the transition matrices
$\bm{P_{2}}$ during the frame block. Since $P^n$ means the state
transition probability after n frames, we can calculate the
probability of state transition after a frame block when we use
$\bm{P_{1}}^{N_{L}}$ and $\bm{P_{2}}^{N_{L}}$. According to proposed
scheme, when the femto AP determines operation mode, the femto AP can't
have more than $\beta$ packets in the queue if the power mode becomes
mode1 and the queue length have to be less than $\alpha$ when the
power mode becomes mode2. Therefore, if $q \ge \beta$, $(1,q,c)\notin
(M_{N}, Q_{N \times N_{L}}, C_{N \times N_{L}})$ and if $q < \alpha$,
$(2,q,c)\notin (M_{N}, Q_{N \times N_{L}}, C_{N \times N_{L}})$.

The transition probability from $(1,q,c)$ to $(1,q',c')$ is expressed as
%===========================================================================
\begin{equation}
\bm{P}_{(1,q,c),(1,q',c')} = {\bm{P_{1}}^{N_{L}}}_{(q,c),(q',c')}. 
\label{fgdfvbvsdf}
\end{equation}
%========================================================================
When the mode changes from 1 to 2, the channel state also changes. Since the transit power increases, the channel state is added by one except the case that the channel state is already the maximum number 9. Therefore, the transition probability is described as follows:
%==========================================================
\begin{numcases}{\bm{P}_{(1,q,c),(2,q',c')}=} \nonumber
{\bm{P_{1}}^{N_{L}}}_{(q,c),(q',c'-1)},  & if $1<c'<9$ \\ 
0, & if $c'=1$ \label{vedzd}
\end{numcases}
%=========================================================
%=============================================================================
\begin{equation}
\bm{P}_{(1,q,c),(2,q',9)}= {\bm{P_{1}}^{N_{L}}}_{(q,c),(q',8)} + {\bm{P_{1}}^{N_{L}}}_{(q,c),(q',9)}.
\label{bdfvdf}
\end{equation}
%=============================================================================
Simillary, the transition probability from $(2,q,c)$ to $(2,q',c')$ is expressed as
%==========================================================================
\begin{equation}
\bm{P}_{(2,q,c),(2,q',c')} = {\bm{P_{2}}^{N_{L}}}_{(q,c),(q',c')}.
\label{adgbvvde}
\end{equation}
%==========================================================================
When the mode is changed from 1 to 2, the femto AP increases its
transmission power by 3 dB. Thus, the transition probability from (2,q,c) to (1,q',c') is represented as follows by considering the channel state changing according to the mode transition.
%==========================================================
\begin{numcases}{\bm{P}_{(2,q,c),(1,q',c')}=} \nonumber
{{\bm{P_{2}}^{N_{L}}}_{(q,c),(q',c'+1)}},  & if $1<c'<9$ \\ 
0, & if $c'=9$
\label{aaeeebbb}
\end{numcases}
%=========================================================
%=============================================================================
\begin{equation}
\bm{P}_{(2,q,c),(1,q',1)}= {\bm{P_{1}}^{N_{L}}}_{(q,c),(q',1)} + {\bm{P_{1}}^{N_{L}}}_{(q,c),(q',2)}
\label{bdfvdfddfd}
\end{equation}
%=============================================================================
By using the equations from (\ref{fgdfvbvsdf}) to
(\ref{bdfvdfddfd}), the state transition matrix is defined as
Eq.(\ref{tretwr}). Since the Markov chain which is defined by the
transition matrix $\bm{P}$ is finite, homogeneous, and irreducible,
the markov has an unique stationary distribution $\pi$ \cite{markov}.
%========================================================
% \begin{figure*}[!t]

% \normalsize
\small
%===========================================================================
\begin{equation}
\bm{P} = 
\begin{bmatrix}
P_{(1,0,1),(1,0,1)} & \cdots&P_{(1,0,1),(1,0,9)} & \cdots & P_{(1,0,1),(1,\beta -1,9)}& P_{(1,0,1),(2,\alpha,1)}& \cdots & P_{(1,0,1),(2,Q_{L},9)}\\
\vdots & \ddots & \vdots & \ddots & \vdots & \vdots &\ddots &\vdots \\
P_{(1,0,9),(1,0,1)}& \cdots& P_{(1,0,9),(1,0,9)}& \cdots& P_{(1,0,9),(1,\beta -1,9)}& P_{(1,0,9),(2,\alpha,1)}&\cdots&P_{(1,0,9),(2,Q_{L},9)}\\
\vdots & \ddots & \vdots & \ddots & \vdots & \vdots &\ddots &\vdots \\
P_{(1,\beta -1,9),(1,0,1)} & \cdots& P_{(1,\beta -1,9),(1,0,9)}& \cdots& P_{(1,\beta -1,9),(1,\beta -1,9)}& P_{(1,\beta -1,9),(2,\alpha,1)}&\cdots&P_{(1,\beta -1,9),(2,Q_{L},9)}\\
P_{(2,\alpha,1),(1,0,1)} & \cdots& P_{(2,\alpha,1),(1,0,9)}& \cdots& P_{(2,\alpha,1),(1,\beta -1,9)}& P_{(2,\alpha,1),(2,\alpha,1)}&\cdots&P_{(2,\alpha,1),(2,Q_{L},9)}\\
\vdots & \ddots & \vdots & \ddots & \vdots & \vdots &\ddots &\vdots \\
P_{(2,Q_{L},9),(1,0,1)} & \cdots& P_{(2,Q_{L},9),(1,0,9)}& \cdots& P_{(2,Q_{L},9),(1,\beta -1,9)}& P_{(2,Q_{L},9),(2,\alpha,1)}&\cdots&P_{(2,Q_{L},9),(2,Q_{L},9)}\\
\label{tretwr}
\end{bmatrix}
\end{equation}
%===========================================================================
% \hrulefill
% \vspace*{4pt}
% \end{figure*}
%========================================================
\normalsize
Then, the stationary distribution $\pi$ satisfies the equality
%=============================================================================
\begin{equation}
\pi = \pi \bm{P}.
\end{equation}
%=============================================================================
If we know the stationary distribution, we could estimate the system
performance. Fortunately, the stationary probability can be calculated
by using computer programs.

In order to analyze the system performance by using the stationary
distribution $\pi$, we shold split the $\pi$ according to the operation mode
because the state transition during a flame block is dependent on the
mode. Let $\pi_{1}$ and $\pi_{2}$ denote the fraction of the $\pi$ for
mode1 and mode2, respectively. Then, the $\pi_{1}$ is defined for all
$(Q_{n} , C_{n})$ as follows:
%================================================================
\begin{equation}
\pi_{1} = [{\pi_{1}}_{(0,1)},\cdots ,{\pi_{1}}_{(0,9)},\cdots ,{\pi_{1}}_{(Q_{L},1)},\cdots ,{\pi_{1}}_{(Q_{L},9)}],
\end{equation}
%================================================================
where the ${\pi_{1}}_{(q,c)}$ can be expressed as
%==========================================================
\begin{numcases}{ {\pi_{1}}_{(q,c)}= } \nonumber
   \pi_{(1,q,c)},  & for $q < \beta$ \\
    0.  & for otherwise  \label{pa}
\end{numcases}
%=========================================================
Similary, the $\pi_{2}$ is also described as
%================================================================
\begin{equation}
\pi_{1} = [{\pi_{2}}_{(0,1)},\cdots ,{\pi_{2}}_{(0,9)},\cdots ,{\pi_{2}}_{(Q_{L},1)},\cdots ,{\pi_{2}}_{(Q_{L},9)}] \label{thfgw}
\end{equation}
%================================================================
%==========================================================
\begin{numcases}{ {\pi_{2}}_{(q,c)}= } \nonumber
    0,  & for $q < \alpha$ \\ 
    \pi_{(2,q,c)}.  & for otherwise  \label{gefdvcvd}
\end{numcases}
%=========================================================


We can exploit the $\pi_{1}$ and $\pi_{2}$ to calculate the packet
drop probability due to the overflow of the queue. The packet drops
can appear at every frame although the stationary distribution is
calculated for frame blocks. Therefore, when we calculate the expected
number of dropped packets, we have to sum the dropped packets during a
frame block. Let $D_{V(q,c)}$ denote the expected number of dropped
packets at state (q,c) and $D_V$ denote the vector that consists of
the $D_{V(q,c)}$. Then, the $D_V$ is represented as
%================================================================
\begin{equation}
D_{V} = [D_{V(0,1)},\cdots ,D_{V(0,9)},\cdots ,D_{V(Q_{L},1)},\cdots ,D_{V(Q_{L},9)}]^{T} \label{thfgw}
\end{equation}
%================================================================
and the $D_{V(q,c)}$ is calculated as Eq.(\ref{dagaggad}).
%========================================================
% \begin{figure*}[!t]

% \normalsize

\begin{equation}
\label{dagaggad} 
D_{V(q,c)} = \sum_{a\ge 0}[max\{0,a-Q_{L}+max\{0,q-MCS(c)\} \}\times P(A=a)] 
\end{equation}

% \hrulefill
% \vspace*{4pt}

% \end{figure*}
%========================================================
By using the vector $D_{V}$, we can obtain the expected number of the dropped packets during one frame block as 
%=========================================================
\begin{equation}
E\{Drop\}=\sum^{2}_{j=1} \pi_{j} \times ( \sum^{N_{L}-1}_{i=0}P^{i}_{j} ) \times D_{V},
\end{equation}
%=========================================================
since, during the frame block, the stationary probabilities $\pi_{1}$
and $\pi_{2}$ are varied with transition matrix $P_{1}$ and $P_{2}$,
respectively. Finally, we can derive the packet dropping probability
and the expected throughput as follows, respectively.
%========================================================
\begin{equation}
\label{fadfa}
P_{d}= \frac{E\{Drop\}}{\lambda N_{L}} 
\end{equation}
%=========================================================
\begin{eqnarray}\nonumber
E\{Throughput\}&=&(\lambda )(1-P_{d})\\
               &=& \lambda - \frac{E\{Drop\}}{N_L}
\label{ogkvdie}
\end{eqnarray}
%========================================================


\hfill

\section{Numerical Results}\label{sec:numerical-results}
In previous section, we make a system model by using FSMC model. In
this section we explain the numerical results. The parameters, which
are used for the numerical results, are defined in
Table. \ref{parameters}.

\begin{table}[!t]
\renewcommand{\arraystretch}{1.2}
\caption{Parameters for numerical results}
\label{parameters}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Parameter& Value \\
\hline \hline
 Average SINR $\Bar{\gamma}$& 20 dB\\ 
\hline
 Frame block length $N_{L}$& 5 frames\\
\hline
 Maximum queue length $Q_L$& 100 packets\\
\hline
 Nakagami parameter m& 1\\
\hline
 Frame duration $T_f$& 5 ms\\
\hline
 Doppler frequency $f_d$& 4 Hz\\
\hline
\end{tabular}
\end{center}
\end{table}
%================================================================
\begin{figure}[!t]
\centering{\includegraphics[width=0.7\textwidth]{fig/results3}}
\caption{Average throughput vs traffic density} \label{result1}
\end{figure}
%================================================================
%================================================================
\begin{figure}[!t]
\centering{\includegraphics[width=0.7\textwidth]{fig/results2}}
\caption{Probability of mode 1 vs traffic density} \label{result2}
\end{figure}
%================================================================

Fig. \ref{result1} represents the average throughput vs. traffic
density. It is shown that our proposed scheme has almost the same data
rate as the conventional scheme which uses constant transmission
power. It is because the femto BS reduce his transmission power only
if the queue length is below threshold and the threshold is
low. Therefore, we can expect that the throughput of the femto cell
doesn't decrease even if the femto AP uses the proposed scheme.

We show the probability of the mode 1 so that we could get results for
power consumption of the femto AP. Since the femto AP uses half power
to transmit a frame in mode1, the probability of the mode 1 means power
saving factor. Fig. \ref{result2} shows the probability of mode1 as
traffic density increases. As you can see, the probability decreases as the
traffic density increases. Therefore, we can make conclusions that the
proposed scheme can reduce its transmission power without any
throughput loss since the proposed scheme changes the transmission
power according to traffic density and the proposed scheme can reduce
interference to neighbor cells since it reduces its transmission
power.


\hfill


\section{Summary}\label{sec:conclusions-1}
In this chapter, we have suggested a solution for femto cell
interference issues. The proposed scheme could reduce the downlink
interference from a femto AP to neighbor cell users without throughput
loss because it controls a femto AP's transmission power adaptively
according to the traffic density of the femto AP. Moreover, based on
the FSMC model \cite{LZ05QAM}, we analyze the performance of proposed
system. Since our proposed scheme changes the transmission power of the
femto AP every $N_L$ frames interval, we introduce a fame block concept
and construct FSMC model with the frame block. Therefore, we can
analyze our proposed scheme though the scheme changes the transmission
power according to the traffic load of the femto AP.

There are some remaining challenges for femto AP's power
control. Since we just show the power reduction of femto AP, we should
analyze throughput enhancement of neighbor nodes. Moreover, we should
study more powerful scheme than our proposed one.


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